Basic Math Examples

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 x + 8}{x^{2} - 5 x}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $2$ out of $2 x + 8$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x) + 8}{x^{2} - 5 x}$

Step 1.1.2

Factor $2$ out of $8$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 x + 2 \cdot 4}{x^{2} - 5 x}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x^{2} - 5 x}$

Step 1.2

Factor $x$ out of $x^{2} - 5 x$ .

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Step 1.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x \cdot x - 5 x}$

Step 1.2.2

Factor $x$ out of $- 5 x$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x \cdot x + x \cdot - 5}$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} - 5^{2}}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{(x + 5) (x - 5)}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{(x + 5) (x - 5)}{x^{2} - 4^{2}} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x^{2} - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4

Simplify terms.

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Step 4.1

Factor $x$ out of $x^{2} - 4 x$ .

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Step 4.1.1

Factor $x$ out of $x^{2}$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x \cdot x - 4 x}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.1.2

Factor $x$ out of $- 4 x$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x \cdot x + x \cdot - 4}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 x + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.2

Factor $2$ out of $2 x + 10$ .

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Step 4.2.1

Factor $2$ out of $2 x$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 (x) + 10} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.2.2

Factor $2$ out of $10$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 x + 2 \cdot 5} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.2.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 (x + 5)} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.3

Cancel the common factor of $x + 5$ .

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Step 4.3.1

Factor $x + 5$ out of $2 (x + 5)$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{(x + 5) \cdot 2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.3.2

Cancel the common factor.

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{(x + 5) \cdot 2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.3.3

Rewrite the expression.

$\frac{x - 5}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.4

Cancel the common factor of $x - 4$ .

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Step 4.4.1

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{x (x - 4)}{2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.4.2

Factor $x - 4$ out of $x (x - 4)$ .

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{(x - 4) x}{2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.4.3

Cancel the common factor.

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{(x - 4) x}{2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.4.4

Rewrite the expression.

$\frac{x - 5}{x + 4} \cdot \frac{x}{2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.5

Multiply $\frac{x - 5}{x + 4}$ by $\frac{x}{2}$ .

$\frac{(x - 5) x}{(x + 4) \cdot 2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.6

Cancel the common factor of $x (x - 5)$ .

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Step 4.6.1

Factor $x (x - 5)$ out of $(x - 5) x$ .

$\frac{x (x - 5) (1)}{(x + 4) \cdot 2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.6.2

Cancel the common factor.

$\frac{x (x - 5) \cdot 1}{(x + 4) \cdot 2} \cdot \frac{2 (x + 4)}{x (x - 5)}$

Step 4.6.3

Rewrite the expression.

$\frac{1}{(x + 4) \cdot 2} \cdot (2 (x + 4))$

Step 4.7

Cancel the common factor of $2 (x + 4)$ .

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Step 4.7.1

Factor $2 (x + 4)$ out of $(x + 4) \cdot 2$ .

$\frac{1}{2 \cdot (x + 4) (1)} \cdot (2 (x + 4))$

Step 4.7.2

Cancel the common factor.

$\frac{1}{2 \cdot (x + 4) \cdot 1} \cdot (2 (x + 4))$

Step 4.7.3

Rewrite the expression.

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